9.2: Temperature Dependence of ΔG
\[ \left(\dfrac{\partial G}{\partial T} \right)_{P,\{n_i\}}=-S \nonumber \]
Let’s analyze the first coefficient that gives the dependence of the Gibbs energy on temperature. Since this coefficient is equal to \(-S\) and the entropy is always positive, \(G\) must decrease when \(T\) increases at constant \(P\) and \(\{n_i\}\), and vice versa.
If we replace this coefficient for \(-S\) in the Gibbs equation, Equation 9.1.3 , we obtain:
\[ \Delta G = \Delta H + T \left(\dfrac{\partial \Delta G}{\partial T} \right)_{P,\{n_i\}}, \label{9.2.1} \]
and since Equation \ref{9.2.1} includes both \(\Delta G\) and its partial derivative with respect to temperature \(\left(\dfrac{\partial \Delta G}{\partial T} \right)_{P,\{n_i\}}\) we need to rearrange it to include the temperature derivative only. To do so, we can start by evaluating the partial derivative of \(\left( \dfrac{\Delta G}{T} \right)\) using the chain rule:
\[ \left[ \dfrac{\partial\left( \dfrac{\Delta G}{T} \right)}{\partial T} \right]_{P,\{n_i\}} = \dfrac{1}{T} \left(\dfrac{\partial \Delta G}{\partial T} \right)_{P,\{n_i\}} - \dfrac{1}{T^2}\Delta G, \label{9.2.2} \]
which, replacing \(\Delta G\) from Equation \ref{9.2.1} into Equation \ref{9.2.2}, becomes:
\[ \begin{equation} \begin{aligned} \left[ \dfrac{\partial\left( \dfrac{\Delta G}{T} \right)}{\partial T} \right]_{P,\{n_i\}} &= \dfrac{1}{T} \left(\dfrac{\partial \Delta G}{\partial T} \right)_{P,\{n_i\}} - \dfrac{1}{T^2} \left[ \Delta H + T \left(\dfrac{\partial \Delta G}{\partial T} \right)_{P,\{n_i\}} \right] \\ &= \dfrac{1}{T} \left(\dfrac{\partial \Delta G}{\partial T} \right)_{P,\{n_i\}}- \dfrac{\Delta H}{T^2}-\dfrac{1}{T} \left(\dfrac{\partial \Delta G}{\partial T} \right)_{P,\{n_i\}}, \end{aligned} \end{equation} \label{9.2.3} \]
which simplifies to:
\[ \begin{equation} \begin{aligned} \left[ \dfrac{\partial\left( \dfrac{\Delta G}{T} \right)}{\partial T} \right]_{P,\{n_i\}} &= - \dfrac{\Delta H}{T^2}. \end{aligned} \end{equation} \label{9.2.4} \]
Equation \ref{9.2.4} is known as the Gibbs–Helmholtz equation , and is useful in its integrated form to calculate the Gibbs free energy for a chemical reaction at any temperature \(T\) by knowing just the standard Gibbs free energy of formation and the standard enthalpy of formation for the individual species, which are usually reported at \(T=298\;\text{K}\). The integration is performed as follows:
\[ \begin{equation} \begin{aligned} \int_{T_i=298 \;\text{K}}^{T_f=T} \dfrac{\partial\left( \dfrac{\Delta_{\text{rxn}} G}{T} \right)}{\partial T} &= - \int_{T_i=298 \;\text{K}}^{T_f=T} \dfrac{\Delta_{\text{rxn}} H}{T^2} \\ \\ \dfrac{\Delta_{\text{rxn}} G^{-\kern-6pt{\ominus}\kern-6pt-}(T)}{T} &= \dfrac{\Delta_{\text{rxn}} G^{-\kern-6pt{\ominus}\kern-6pt-}}{298 \;\text{K}} + \Delta_{\text{rxn}} H^{-\kern-6pt{\ominus}\kern-6pt-}\left( \dfrac{1}{T^2} -\dfrac{1}{(298 \;\text{K})^2} \right), \end{aligned} \end{equation} \label{9.2.5} \]
giving the integrated Gibbs–Helmholtz equation:
\[ \dfrac{\Delta_{\text{rxn}} G^{-\kern-6pt{\ominus}\kern-6pt-}(T)}{T} = \dfrac{\sum_i \nu_i \Delta_{\text{f}} G_i^{-\kern-6pt{\ominus}\kern-6pt-}}{298 \;\text{K}} + \sum_i \nu_i \Delta_{\text{f}} H_i^{-\kern-6pt{\ominus}\kern-6pt-}\left( \dfrac{1}{T^2} -\dfrac{1}{(298 \;\text{K})^2} \right) \label{9.2.6} \]